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Click on their Followers and begin Following only users that aren't Private. You will only be able to follow 30 per hour.

Wait an hour and repeat, do this about 3 times so you've given the users you've followed time to follow you back.

Click on your Followers and begin to UNFOLLOW anyone who hasn't followed you back. This makes room for you to follow more users.

Keep in mind you can only follow upto 7,500 people, so try to Unfollow as often as possible to stay away from that Follower Cap.

You'll begin to see immediate results, people will follow you right away.

The last thing you might want to do, and it's not required is use an app called: Social Blade

Be sure to Add your Page and start tracking your daily Followers. The app will show you how many Followers you are Gaining or Losing everyday. Hint: You can also track your competition!

You might also consider buying some Followers to show Social Proof. People tend to follow pages that Look Popular: You can use sites like: socialproof.xyz seoclerks.com or fiverr.com to get cheap followers.
If you have money to spend then you should be buying Shoutouts on your Niche sites. It's expensive but a really fast way to gain followers using money.

At some point, organic followers usually kick in to propel your account to superstardom, but the follow/unfollow method provides a quick, easy, and free way to get your Instagram account off the ground.
Topics that are covered in the guide include:

Good advice. Here's an article that walks you through the follow/unfollow method: http://reviewsofcoolstuff.com/social-media/how-i-got-my-first-5000-instagram-followers-in-about-a-month-gaining-500-1k-followersweek/